Ratio Working Paper No. 345: Regularized Conditional Estimators of Unit Inefficiency in Stochastic Frontier Analysis, with Application to Electricity Distribution Market

PublicationWorking paper
Electricity Distribution, Productivity, Regularized Posterior Likelihood, Stochastic Frontier Analysis
Ratio Working Paper No. 345


The practical value of Stochastic Frontier Analysis (SFA) is positively related to the level of accuracy at which it estimates unit-specific inefficiencies. Conventional SFA unit inefficiency estimation is based on the mean/mode of the inefficiency, conditioned on the estimated composite error. This approach shrinks the inefficiency towards its mean/mode, which generates a distribution that is different from the distribution of the unconditional inefficiency; thus, the accuracy of the estimated inefficiency is negatively correlated with the distance the inefficiency is located from its mean/mode. We propose a regularized estimator based on Bayesian risk (expected loss) that restricts the unit inefficiency to satisfy the underlying theoretical mean and variation assumptions. We analytically investigate some properties of the maximum a posteriori probability estimator under mild assumptions and derive a regularized conditional mode estimator for three different inefficiency densities commonly used in SFA applications. Extensive simulations show that, under common empirical situations, e.g., regarding sample size and signal-to-noise ratio, the regularized estimator outperforms the conventional (unregularized) approach when the inefficiency is greater than its mean/mode. With real data from electricity distribution sector in Sweden, we demonstrate that the conventional conditional estimators and our regularized conditional estimators give substantially different results for highly inefficient companies.

Zeebari, Z., Månsson, K., Sjölander, P. & Söderberg, M. (2021). Regularized Conditional Estimators of Unit Inefficiency in Stochastic Frontier Analysis, with Application to Electricity Distribution Market. Ratio Working Paper No. 345. Stockholm: Ratio.

Similar content

Working Paper no. 328 Wholesale firms: A catalyst for Swedish exports?
Working paperPublication
Daunfeldt, S-O., Engberg, E., Halvarsson, D., Kokko, A. & Tingvall, P.
Publication year


Published in

Ratio Working Paper


This paper examines the role of wholesale firms as facilitators of exports for small and medium-sized Swedish businesses. Our findings suggest that wholesale firms do facilitate access to difficult markets located outside Europe. For exports of a particular good to a given market, we observe a positive correlation between the export volumes of wholesale and manufacturing firms. Finally, we present evidence that supports a prediction from recent trade models with differentiated firms, namely that wholesale firms can facilitate exports for firms that are not themselves capable of direct exports.

Show more